Counting curves of any genus on P^2_6

TL;DR

This paper presents a new formula for counting algebraic curves of any genus on the blown-up projective plane P^2_6, extending previous formulas and including tangency conditions.

Contribution

It introduces a comprehensive formula for degrees of varieties parameterizing curves of any genus on P^2_6, generalizing prior results and providing an alternative to Vakil's approach.

Findings

Formula computes degrees for curves with various genus and divisor classes.

Includes tangency conditions to a fixed exceptional divisor.

Special cases recover known counts on P^2_q and (P^1)^2.

Abstract

We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the varieties parameterizing curves on P^2_6 which additionally satisfy certain tangency conditions to a fixed exceptional divisor on P^2_6. Our formula contains as special cases the degrees of the analogous varieties parameterizing curves on P^2_q, for q=1,...,5, and on the quadric (P^1)^2. It is an extension of the Caporaso-Harris formula counting curves of any degree and genus in the projective plane, and it can be viewed as an alternative to the Vakil formula, which gives an indirect way to count curves of any genus on P^2_6.